Problem: $ B = \left[\begin{array}{rr}3 & 5 \\ 5 & 2\end{array}\right]$ $ D = \left[\begin{array}{rrr}-2 & 3 & 2 \\ 3 & -2 & 5\end{array}\right]$ What is $ B D$ ?
Explanation: Because $ B$ has dimensions $(2\times2)$ and $ D$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ B D = \left[\begin{array}{rr}{3} & {5} \\ {5} & {2}\end{array}\right] \left[\begin{array}{rrr}{-2} & \color{#DF0030}{3} & \color{#9D38BD}{2} \\ {3} & \color{#DF0030}{-2} & \color{#9D38BD}{5}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{5}\cdot{3} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{5}\cdot{3} & ? & ? \\ {5}\cdot{-2}+{2}\cdot{3} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{5}\cdot{3} & {3}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{-2} & ? \\ {5}\cdot{-2}+{2}\cdot{3} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{3}\cdot{-2}+{5}\cdot{3} & {3}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{-2} & {3}\cdot\color{#9D38BD}{2}+{5}\cdot\color{#9D38BD}{5} \\ {5}\cdot{-2}+{2}\cdot{3} & {5}\cdot\color{#DF0030}{3}+{2}\cdot\color{#DF0030}{-2} & {5}\cdot\color{#9D38BD}{2}+{2}\cdot\color{#9D38BD}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}9 & -1 & 31 \\ -4 & 11 & 20\end{array}\right] $